Question

which linear inequality is represented by the graph?
\\( y\leq \frac{1}{2}x+2 \\)
\\( y\geq \frac{1}{2}x+2 \\)
\\( y\leq \frac{1}{3}x+2 \\)
\\( y\geq \frac{1}{3}x+2 \\)

Explanation:

Step1: Find the slope of the line

The line passes through (0, 2) and (4, 4). The slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 2}{4 - 0}=\frac{2}{4}=\frac{1}{2} \). So the equation of the line is \( y=\frac{1}{2}x + 2 \) (since the y - intercept is 2).

Step2: Determine the inequality symbol

The shaded region is above the line, and the line is solid (so the inequality includes equality). So the inequality is \( y\geq\frac{1}{2}x + 2 \).

Answer:

B. \( y\geq\frac{1}{2}x + 2 \)